Can any object with positive mechanical energy collide earth?

Object with positive mechanical energy in an attractive force field moves along a hyperbolic path and whose energy is zero, moves along parabolic path...isn't it?...But why?...Can't any object with ve energy move along the line joining centre of force?...If its radial component of kinetic energy is greater than effective potential energy? Then? If I'm right then spacial object can collide earth....
Please ellaborate conceptually and help me out...
Thanks.

The sum of Potential and Kinetic energy (total energy), on impact can be greater than Zero. All that's necessary is that it was given a big enough kick, whilst way out in space, in the direction of Earth, so that it hits the Earth with more than the escape velocity for an object being launched from the Earth's surface. So its KE (always positive) will be greater than its Potential Energy (negative).

Couldn't get it clearly. Can you please explain why the paths of objects are hyperbolic, parabolic and elliptic for energies positive, zero and negative respectively?

I assume you know the mathematical form of the different conic sections. The 'reason' they turn out to be the paths of ideal orbits is very mathematical and they are what you get by solving the equation of motion of a body under the influence of an inverse square law force. The conic sections are not arm waving shapes and can't really be derived in an arm waving way. The Maths does it perfectly in very few lines.
I got a lot of google hits when I searched with terms like orbit, gravity, hyperbola etc.This one may be interesting.

Well...finally after a google search for ''how did kepler derive his first law'', I could find how kepler did it.... Although I couldn't understand totally....but I'm satisfied because he didn't use the Energy condition....By Brahe's observation data and his trial and error, he could successfully do it...
Here is the website...
http://math.berkeley.edu/~robin/Kepler/

Dear old Kepler didn't use the Energy method because no one was aware of the Laws of Gravitation at the time. He was just fitting the simplest mathematical model to the observations - really smart. I am left breathless by the early astronomers did with their very basic observation techniques (+ lots of total dedication).

Ya...absolutely...but one thing is that Kepler just explained the nature of orbits. He didn't explain why the nature is so. Now a days we have lots of mathematics for that. But these mathematics could not satisfy me...Hope they will...

Ya...absolutely...but one thing is that Kepler just explained the nature of orbits. He didn't explain why the nature is so. Now a days we have lots of mathematics for that. But these mathematics could not satisfy me...Hope they will...

Our math does a wonderful job of showing HOW many phenomena work, but it will never show WHY they work. The answer to a "why" question is always another "why" question.

You can say that the mathematical laws of motion show why planets have elliptical orbits but then you have to ask why to those laws work? You can get to general relativity and say that they work because of curved space-time, but you have to ask, why is space-time curved? The bottom question in the chain is always another "why" question.

Our math does a wonderful job of showing HOW many phenomena work, but it will never show WHY they work. The answer to a "why" question is always another "why" question.

You can say that the mathematical laws of motion show why planets have elliptical orbits but then you have to ask why to those laws work? You can get to general relativity and say that they work because of curved space-time, but you have to ask, why is space-time curved? The bottom question in the chain is always another "why" question.

Totally agree with you...There is always a ''Why?''. But some times some answer to 'why' satisfies us and we don't ask further 'why'.
Although I have not read Relativity...so I think I have to wait until I gather sufficient knowledge...